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Physical Review A

Tunneling of squeezed states in asymmetrical double-well potentials

Daniela Mugnai, Anedio Ranfagni, M. Montagna, Oreste Pilla, G. Viliani, Maurizio Cetica

Published Aug 1, 1988• 16 citations
Publisher

Abstract

Tunneling of initially squeezed wave packets in asymmetrical double-well potentials is studied; the fractal nature of the trajectory, previously found in the case of symmetric potentials, is re- obtained.
DOI
10.1103/physreva.38.2182

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Journal of Physics A Mathematical and General•Dec 21, 1986•7 citations
Methods previously developed for the case of a symmetric double-well potential are here extended to study tunneling in an asymmetric, piecewise harmonic, double-well potential. The approach involves the decomposition of the real stationary states into complex right- and left-moving states and the determination of exact transmission coefficients for the latter. Numerical solutions for a selection of representative parameter values are obtained for the time evolution of states which correspond initially to a Gaussian wave packet localized in the upper well. Both resonant and nonresonant cases are studied and the significance of the transmission coefficient is established for each.
The motion of the center of a wave packet localized in one of the wells of an asymmetric double-well potential is studied. The asymmetric potential is obtained from a solvable symmetric potential using the Gel'fand–Levitan formalism, by keeping the energy eigenvalues unchanged, but changing the normalization of the ground-state wave function. While in a symmetric double well the wave packet tunnels back and forth between the two wells, for an asymmetric well, in general, tunneling does not take place. This is independent of the initial location of the wave packet, i.e., whether it is centered at the minimum of the shallower or of the deeper well. However, when such a system is coupled to a dissipative force, then the tunneling becomes possible. For instance it is shown that if, as a model of dissipative coupling, one chooses Gisin's nonlinear evolution equation, then the center of the wave packet ends up in the deeper well (decay of false vacuum). This result depends on the particular model of dissipation, for instance, an optical potential model yields a different result.
WKB calculations have revealed that tunneling dynamics of localized wave packets in simple potential wells depend sensitively on the degree of squeezing of the wave function. Squeezed states are shown to tunnel more slowly in general than eigenstates of the well and states which are initially localized near the bottom of a well have much lower tunneling rates than states localized at the sides of the well. The dependence of tunneling time on energy also exhibits steps related to quasi-bound-state energies in a complex way. Additionally, modulations in the tunneling probability correlate with the motion of the squeezed wave packet in the well. These results have implications both for the dynamics of squeezed radiation fields and squeezed matter states, and examples of applications to controlled vibrational-state chemistry, ultralow-noise measurements of weak signals with Josephson junctions, and optical crystals are discussed.